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Extremal Oriented Graphs and Erd˝Os-Hajnal Conjecture

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Extremal Oriented Graphs and Erd˝Os-Hajnal Conjecture EXTREMAL ORIENTED GRAPHS AND ERDOS-HAJNAL} CONJECTURE by Payam Valadkhan B.Sc., Sharif University of Technology, 2007 a Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the School of Computing Science c Payam Valadkhan 2009 SIMON FRASER UNIVERSITY Summer 2009 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Payam Valaclkhan Degree: N{aster o[ Science Title of Thesis: Extrernal OrierrteclGraphs and Erd6s-Hajnal Conjecture Examining Committee: Dr. RarrreshKrisirnamurti Cirair Dr. Gd,bor Tardos, Professor',Compnting Science Sirnori Fra,serUniversity Seriior Supervisor Dr. Arrrlrei Bulat,ov. Associate Professor, Ooniputing Scierrce SirrronF'raser L rriversitv Superrvisor Dr. LaclislarvStacho. Associate Professor. N,{at}renriit- ics Simorr Fraser University SFU Exanriner Date Approved: /ll,tnp vl tll- 4'1-lq| ./ ilu ii^rl-)r,C,') ,/ !,t Declaration of Partial Copyright Licence The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the “Institutional Repository” link of the SFU Library website <www.lib.sfu.ca> at: <http://ir.lib.sfu.ca/handle/1892/112>) and, without changing the content, to translate the thesis/project or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without the author’s written permission. Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence. While licensing SFU to permit the above uses, the author retains copyright in the thesis, project or extended essays, including the right to change the work for subsequent purposes, including editing and publishing the work in whole or in part, and licensing other parties, as the author may desire. The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive. Simon Fraser University Library Burnaby, BC, Canada Last revision: Spring 09 Abstract For a (finite or infinite) family L of oriented graphs, a new parameter called the compress- ibility number of L and denoted by z(L) is defined. The main motivation is the application of this parameter in a special case of Tur´an-type extremal problems for digraphs, in which it plays the role of chromatic number in the classic extremal problems. We estimate this parameter for some special group of oriented graphs. Determining this parameter, in the most explicit possible form, for oriented graphs with bounded oriented coloring number (planar graph in particular) leads us to the famous Erd}os-Hajnal conjecture. iii Contents Approval ii Abstract iii Contents iv 1 Introduction 1 1.1 Background and old results . .1 1.2 New results and the thesis structure . .5 1.3 Notations . .6 2 The nature of compressibility number 7 2.1 The existence . .7 2.2 Oriented Erd}os-Stone-Simonovits . .7 3 Compressibility number for some oriented graphs 13 3.1 Sparse graphs . 13 3.2 Dense graphs . 16 4 Relationship with Erd}os-Hajnal conjecture 19 5 Some oriented Erd}os-Hajnal problems 25 5.1 Partitioning restrictions . 25 5.2 A constructive method . 27 Bibliography 30 iv Chapter 1 Introduction 1.1 Background and old results By digraph we mean a directed graph which contains no loop and no multi-arcs: two or more arcs with the same direction between a pair of vertices. But it can have two opposite arcs between a pair of vertices, which are called symmetric arcs. For any digraph H we denote by V (H) and E(H), the set of vertices and arcs of H, respectively. The same definitions apply when replacing the notion digraph and arc with simple graph and edge, respectively. From now on by graph we mean simple graph. For x; y 2 V (H), we write xy−! to indicate that there is an arc in E(H) oriented from x to y. The same notation also represents the arc itself. A digraph H is called a sub-digraph of another digraph H0, and denoted by H ⊆ H0, if V (H) ⊆ V (H0) and E(H) ⊆ E(H0). For S ⊆ V (H0), we say the 0 0 digraph H is called the induced sub-digraph of S in H , and denote by H jS, if V (H) = S and E(H) consists of all members of E(H0) with their endpoints in S. A homomorphic mapping f from a digraph H to another digraph H0 is a mapping f : −−−−−−! V (H) ! V (H0) such that f(x)f(y) whenever xy−!. In this case we write H ! H0 and say H is homomorphic to H0. Should the mapping f be a bijection with its inverse function being also a graph homomorphic mapping, then f is called a graph isomorphic mapping and we say H is isomorphic to H0. We say that H0 contains H if H is isomorphic to a sub-digraph of H0, otherwise we say H0 avoids H or, equivalently, H0 is H-free. 1 CHAPTER 1. INTRODUCTION 2 In their joint paper, [5], Brown and Harary initiated the study of extremal digraphs by considering the digraph analogues of the Paul Tur´an'sclassic extremal theorem [19, 20]: What is the maximum number of arcs in a digraph avoiding a certain forbidden digraph? And what are such extremal digraphs? Their study gives precise answers for the forbidden digraphs with four or less vertices. Later on Brown, Erd}osand Simonovits, in a series of papers [2, 3, 4], studied this problem and its extension to any family (finite or infinite) of forbidden digraphs. Given integer n ≥ 1 and a square matrix A with zero or one along the main diagonal and zero or two anywhere else, in [2], they introduce a method for constructing a digraph A(n) where jV (A(n))j = n. Then they show that for any family (finite or infinite) of forbidden digraphs, there is a matrix A such that for every n ≥ 1, A(n) contains non of the forbidden digraphs and jE(A(n))j is very close to its optimal. Thus, this study ensures the existence of such a family of approximately extremal digraphs and gives some insight into its structure. But it does not answer the digraph version of the Paul Tur´an'sextremal theorem, i.e., a clear estimation for the number of arcs of the extremal digraphs with n vertices. This is due to the unknown order of the matrix A which is used in the construction. Attempts were made to find some estimations for this value. In [4] an algorithm is intro- duced to calculate it. In [16] some upper bounds were proved in terms of some parameters of the forbidden digraphs. These studies revealed the inherent complexity for finding any meaningful estimation of the number of extremal digraphs's arcs. This is in contrast to the undirected version in which a result of Erd}os,Stone and Simonovits [9] gives us an exact asymptotic value for the number of edges of the extremal graphs in terms of the chromatic numbers of the forbidden graphs. The situation becomes more tractable if we limit ourselves to the special case of oriented graphs: a family of digraphs which contain no pair of symmetric arcs. Actually this is suggested by Brown and Harary themselves in the last section of their paper [5] as a possible direction for future works. But this case, which we call the extremal theory of oriented graphs, was not considered in the later works of Brown et al. [2, 3, 4] and other subsequent works [16]. CHAPTER 1. INTRODUCTION 3 The work in the present thesis is along this unstudied direction. We say a oriented graph H is an orientation of a graph G if it is obtained from G by assigning an orientation to each edge of G. If H is acyclic (contains no directed cycle), then we say H is an acyclic orientation of G. An orientation of a complete graph is called tournament. An acyclic tournament is called transitive tournament. From extremal theory's point of view, the chromatic number of a graph G can be re-defined as the smallest number k such that G is homomorphic to the complete graph on k vertices. This definition can be naturally reformulated for oriented graphs by replacing the notion graph with the notion oriented graph, however the term 'complete graph on k vertices' cannot be simply replaced with 'complete oriented graph (=tournament) on k vertices' as we have many such tournaments.
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